Stationary vs measurable limits for large cardinals

Question

This is a follow-up to a question I asked earlier over here. Why are stationary limits so ubiquitous when studying large cardinals?

I have noticed that there appears to be a stronger limit notion that appears sometimes - a "measurable limit". From what I understand, a cardinal $\kappa$ is a measurable limit of cardinals of type X if there is a normal measure on $\kappa$ containing all the cardinals of type X $< \kappa$

From this definition, it is unclear to me why a measurable limit is stronger than a stationary limit. For instance, is a measurable limit something like a stationary limit of stationary limits ? Or something much stronger ?

What would be examples of theorems illustrating why a measurable limit is stronger than a stationary limit ?

Answer 1

If there is a normal measure $U$ on $\kappa$ such that $A_X\in U$, where $A_X$ is the set of $X$-cardinals $\alpha<\kappa$, then not only is $\kappa$ a limit of inaccessiles $\kappa'$ such that $\kappa'$ is a stationary limit of $X$-cardinals, but in fact the set $A_{X'}$ of inaccessible stationary limits of $X$-cardinals ${<\kappa}$ is also in $U$.

For let $j:V\to M$ with $\mathrm{crit}(j)=\kappa$ and $\kappa\in j(A_X)$. Then $M\models$ "$\kappa$ is a stationary limit of $j(X)$-cardinals", because $A_X$ is stationary in $V$ and hence in $M$, and in $M$, $A_X$ is the set of $j(X)$-cardinals $\alpha<\kappa$. It follows that $A_{X'}\in U$.